X-ray radiography and tomography are important methods for a variety of applications, such as e.g. non-destructive investigation of bulk samples, quality inspection of industrial products and non-invasive examination of anatomical structures and tissue regions of interest in the interior of a patient's body, which is because the penetration depth of hard X-ray beams is rather high, which allows for recording sharp projections of the attenuation coefficient. X-ray imaging thereby yields excellent results where highly absorbing anatomical structures such as bones are embedded in a tissue of relatively weakly absorbing material. However, in cases where different kinds of tissue with similar absorption cross-sections are under examination (such as e.g. in mammography or angiography), X-ray absorption contrast is relatively poor. Consequently, since X-ray radiographs or tomographic data sets with sufficient amplitude contrast are often difficult to obtain, differentiating pathological from non-pathological tissue in an absorption radiograph obtained with a current hospital-based X-ray system remains difficult for certain tissue compositions. In particular for medical applications such as mammography, high radiation doses are required to provide a sufficient contrast-to-noise ratio, which severely strains the health of both the patient and the clinical staff.
To overcome these limitations, phase imaging is a promising alternative for radiography of weakly absorbing materials. Several methods to generate radiographic contrast from the phase shift of X-rays passing through a phase object have been investigated. These methods can be classified into interferometric methods, techniques using an analyzer, and free-space propagation methods. All these methods differ vastly in the nature of the signal recorded, the experimental setup, and the requirements on the illuminating radiation. As phase-sensitive imaging techniques require X-rays of high spatial and/or temporal coherence, most of them are either implemented in combination with crystal or multilayer optics, at synchrotron facilities, or they use low-power micro-focus X-ray tubes. As described in “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources” (Nature Physics, vol. 2, num. 4, 2006, pp. 258-261, March 2006, ISSN: 1745-2473) by F. Pfeiffer, T. Weitkamp et al., the required spatial and temporal coherence lengths, ξs and ξt, usually range in the order of about 1 μm. Propagation-based methods can overcome the stringent requirements on the temporal coherence, and—according to Pfeiffer and Weitkamp—have been demonstrated to work well with a broad energy spectrum leading to a temporal coherence length ξt of about 1 nm. However, as shown by these authors, they still require a typical spatial coherence length of ξs≧1 μm, which is currently only available from micro-focus X-ray sources (with correspondingly low power) or synchrotrons. These constraints have, until now, hindered the final breakthrough of phase-sensitive X-ray imaging as a standard method for medical or industrial applications.
The cross section for elastic scattering of hard X-rays in matter, which causes a phase shift of the wave passing through the object of interest, is usually much greater than that for absorption. For example, 17.5-keV X-rays that pass through a 50-μm-thick sheet of biological tissue are attenuated by only a fraction of a percent, while the phase shift in radians is close to π. Recording the X-ray phase shift rather than only the absorption thus has the potential of a substantially increased contrast. A variety of X-ray techniques are employed to detect the phase contrast of a sample, i.e. to convert it into an amplitude contrast in the image plane. Some techniques use the Fresnel diffraction of coherent hard X-rays at the edges of a phase object to significantly improve the visibility of an object in microradiography (see e.g. Snigirev, I. et al., “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation”, Rev. Sci. Instrum. 66 (1995), pp. 5486-5492). In first approximation, the obtained intensity distribution is proportional to the Laplacian of the refractive index distribution such as described in “Observation of microstructure and damage in materials by phase sensitive radiography and tomography” (J. Appl. Phys. 81 (1997), pp. 5878-5886) by P. Cloetens et al. and in certain cases a reconstruction of a phase object from a single micrograph is possible (see Nugent, K. A. et al., “Quantitative phase imaging using hard X-rays”, Phys. Rev. Lett. 77 (1996), pp. 2961-2964). In “Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam” (J. Phys., D. 32 (1999), pp. A145-A151) by P. Cloetens, W. Ludwig et. al., it is described that quantitative information on arbitrary phase objects can be obtained by numerically evaluating series of images acquired with the detector placed at different distances from the sample.
The most sensitive method to measure the phase shifts introduced to a wave front is interferometry. A set-up for a Mach-Zehnder type interferometer operated in the hard X-ray range was introduced in the article “An X-ray interferometer” (Appl. Phys. Lett. 6 (1965), pp. 155-157) by U. Bonse and M. Hart about four decades ago. It consists of three partially transmitting Bragg crystals used as beam splitter and recombining elements. The incoming light is split into two separated branches one of which passes through the sample while the other serves as an unperturbed reference beam. The two beams interfering at the exit of the interferometer give an intensity distribution that represents the difference in optical path and thus—if perfectly aligned—of the phase shift caused by the object. Ando and Hosoya pioneered phase contrast imaging with such a device in the early seventies (see M. Ando and S. Hosoya, in: G. Shinoda, K. Kohra, T. Ichinokawa (Eds.), Proc. 6th Intern. Conf. On X-ray Optics and Microanalysis, “Observation of Antiferromagnetic Domains in Chromium by X-ray Topography”, Univ. of Tokyo Press, Tokyo, 1972, pp. 63-68), and more recent setups have produced large numbers of excellent phase contrast images and computer tomograms, e.g. of biological specimens, such as e.g. described in “Phase-contrast X-ray computed tomography for observing biological specimens and organic materials” (Rev. Sci. Instrum. 66 (1995), pp. 1434-1436) by A. Momose et al. as well as in “Three-dimensional imaging of nerve tissue by X-ray phase-contrast microtomography” (Biophys. J. 76 (1999), pp. 98-102) by F. Beckmann et al. The main technical difficulty lies in the extreme demands on the mechanical stability of the optical components, as the relative positions of the optical components have to be stable within a fraction of a lattice constant, i.e. to sub-Ångström dimensions. Therefore, Bonse-Hart interferometers are very difficult to handle, especially when made big enough to investigate large samples.
A frequently used imaging method for enhancing the contrast of an X-ray radiograph or tomographic image is given by grating-based X-ray differential phase contrast imaging (DPCI), which allows for a simultaneous acquisition of the object absorption as well as a differential phase along a projection line. This technique requires no spatially or temporally coherent sources, is mechanically robust, can be scaled up to large fields of view and provides all the benefits of contrast-enhanced phase-sensitive imaging. Moreover, DPCI is fully compatible with conventional absorption radiography and applicable to X-ray medical imaging, industrial non-destructive testing and all kinds of imaging applications using other types of low-brilliance radiation (such as e.g. neutron radiation). DPCI thus provides valuable additional information usable for contrast enhancement, material composition or dose reduction.
Recently, a group at Paul-Scherrer Institute in Villigen (Switzerland) has shown a simple realization of a new DPCI setup for a Talbot-Lau type hard-X-ray imaging interferometer which can advantageously be applied for medical imaging. In “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources” (Nature Physics, vol. 2, No. 4, 2006, pp. 258-261, March 2006, ISSN: 1745-2473) by F. Pfeiffer, T. Weitkamp et al., a grating interferometer using a differential phase contrast setup is proposed which can be efficiently used to retrieve quantitative phase images with polychromatic X-ray sources of low brilliance. Similarly to equivalent approaches in the visible light or soft X-ray range, it can be shown that two gratings can be used for DPCI using polychromatic X-rays from brilliant synchrotron sources. In the article of Pfeiffer and Weitkamp, it is described how the use of a third grating allows for a successful adaptation of the method to X-ray sources of low brilliance. The proposed setup of these two authors consists of a source grating G0 with period p0, a phase-shifting grating G1 with period p1 (which is placed in downstream direction behind an object O to be imaged and acts as a beam splitter) and an absorber grating G2 with period p2 (see FIGS. 1a and 1b). Source grating G0, which may typically be realized as an arrayed aperture mask with transmitting slits, placed close to the X-ray tube anode, creates an array of individually coherent, but mutually incoherent sources. It effectively allows for the use of relatively large (i.e., square millimeter sized) X-ray sources without compromising on the coherence requirements of the DPCI method. The ratio γ0 of the width of each line source to the source grating period p0 should be small enough to provide sufficient spatial coherence for the DPC image formation process. To be more precisely, for a distance d between gratings G1 and G2 which is given corresponding to the first Talbot distance of d=p12/8λ with λ being the wavelength of the emitted X-ray beam, a spatial coherence length of ξs=λl/γ0p0≧p1 is required, where l denotes the distance between gratings G0 and G1. With a typical value of a few micrometers for period p1, the required spatial coherence length ξs is of the order of about 1 μm, similar to the requirements of other known methods (see “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources” (Nature Physics, vol. 2, num. 4, 2006, pp. 258-261, March 2006, ISSN: 1745-2473) by F. Pfeiffer, T. Weitkamp et al.). It is important to note that even for a setup with only two gratings (G1 and G2) no spatial coherence in the direction parallel to the grating lines is required, in contrast to propagation-based methods. As source grating G0 may contain a large number of individual apertures, each creating a sufficiently coherent virtual line source, standard X-ray generators with source sizes of more than a square millimeter can be used efficiently. To ensure that each line source produced by G0 constructively contributes to the image-formation process, the geometry of the setup should satisfy the condition p0=p2·l/d (see FIGS. 1c-e). It is important to note that the total source size w only determines the final imaging resolution, which is given by wd/l. The arrayed source thus decouples spatial resolution from spatial coherence, and allows the use of X-ray illumination with coherence lengths as small as ξs=λl/w˜10−8 m in both directions, if the corresponding spatial resolution wd/l can be tolerated in the experiment. Finally, on the assumption that a temporal coherence of ξt≧10−9 m is sufficient, it can be deduced that the method of Pfeiffer and Weitkamp (ibid.) requires the smallest minimum coherence volume ξs2·ξt for phase-sensitive imaging if compared with existing techniques. Alternatively to a grating G0, a structured source can be used, as described by J. Baumann et al. in EP 1 803 398 A1. Here, the apertures of G0 are replaced by spatially restricted emission areas of an X-ray source, which is for example represented by a structured anode in an X-ray tube.
The formation process of the resulting DPC image, which is formed by means of phase grating G1 and absorber grating G2, is similar to known methods such as Schlieren imaging or diffraction-enhanced imaging. It essentially relies on the fact that a phase object placed in the X-ray beam path causes a slight deflection of the beam transmitted through the phase object O (see FIG. 1b). The fundamental idea of DPC imaging depends on locally detecting these angular deviations. As described in “Hard X-ray phase tomography with low-brilliance sources” (Physical Review Letters, 2007, vol. 98, Article ID 108105) by F. Pfeiffer, C. Kottler et al., the obtained deflection angle α at phase grating G1, and thus the phases of the intensity oscillations at each pixel position P(x, y), is directly proportional to the local gradient of the object's phase shift and can be quantified by the equation
                                          α            ⁡                          (                              x                ,                y                            )                                =                                                    λ                                  2                  ⁢                  π                                            ·                                                ∂                                      Φ                    ⁡                                          (                                              x                        ,                        y                                            )                                                                                        ∂                  x                                                      =                                          ∫                                  -                  ∞                                                  +                  ∞                                            ⁢                                                                    ∂                                          δ                      ⁡                                              (                                                  x                          ,                          y                          ,                          z                                                )                                                                                                  ∂                    x                                                  ⁢                                                                  ⁢                                  ⅆ                  z                                                                    ,                            (        1        )            
wherein x denotes a transverse direction perpendicular to the interferometer setup's optical axis OA (given by the axis z of the central X-ray beam CXB) and perpendicular to the grating lines of gratings G0, G1 and G2 (given by the y-axis of the three-dimensional Cartesian coordinate system in FIG. 1a), Φ(x, y) represents the phase profile of the incident wave front as a function of the two transverse directions x and y, λ is the wavelength of the incident X-rays, and δ(x, y, z) is the decrement of the real part of the object's refractive index n (x, y, z) from unity, i.e. n (x, y, z)=1−δ(x, y, z)+jβ(x, y, z), with β(x, y, z) denoting the imaginary part of this refractive index and j:=√{square root over (−1)} being the imaginary unit.
For weakly absorbing objects, the detected intensity is a direct measure of the object's local phase gradient ∂Φ(x, y)/∂x. The total phase shift of the object can thus be retrieved by a simple one-dimensional integration along the x-axis. A higher precision of the measurement can be achieved by splitting a single exposure into a set of images taken for different positions of the grating G2. This approach also allows the separation of the DPC signal from other contributions, such as a non-negligible absorption of the object or an already inhomogeneous wave front phase profile before the object. The proposed method of Pfeiffer and Weitkamp is fully compatible with conventional absorption radiography, because it simultaneously yields separate absorption and phase-contrast images such that information is available from both.
A slight deflection angle α, which yields a slight angle of incidence on phase grating G1, results in a local displacement Δx=d·tan (α)≈d·α (for α [rad]<<1) of the interference fringes at the distance d downstream of G1. A local phase gradient ∂Φ(x, y)/∂x caused by the index of refraction of the phase object O can therefore be translated into a local displacement Δx of the interference fringes. Since the direct determination of the exact position of these fringes requires detectors with spatial resolution in the sub-micrometer range, an absorber grating G2, which is realized as a mask of equidistant bars of gold and transmitting slits (see description of FIG. 1e below), is used for determining the average displacement Δx of the fringes within a detector pixel. Absorber grating period p2 thereby equals the periodicity of the undistorted interference pattern.
If absorber grating G2 is stepped perpendicularly to the grating bars (which means in x-direction such as depicted in FIG. 1a) and individual pictures are taken in each position, the measured signal of each pixel becomes an oscillating function of the absorber grating position xg in x-direction. The position of the maximum in this periodic signal is proportional to the average displacement Δx, and thus to the aforementioned local phase gradient ∂φ(x, y)/∂x. Therefore, determining the average displacement shift Δx for each pixel yields an image of the phase gradient. Furthermore, the average over one period of this intensity oscillation is proportional to the transmitted intensity through the object and thus provides the absorption radiograph. Taking advantage of this phase-stepping acquisition mode, both the absorption image and the local phase gradient image can be measured at the same time. For hard X-rays with λ<0.1 nm, deflection angle α is relatively small, typically of the order of a few microradians. In the known setup described above, determination of this deflection angle is achieved by the arrangement formed by phase grating G1 and absorber grating G2. Most simply, it can be thought of as a multi-collimator which translates angular deviations into changes of the locally transmitted intensity that can be detected with a standard imaging detector.
By the way, it has to be noted that the principle of the Talbot-Lau type interferometer is not restricted to line gratings. According to M. Jiang et al. (in: Int. J. Biomed. Imaging, Article ID 827152, Vol. 2008), two-dimensional structured gratings allow to determine a phase gradient ∂Φ(x, y)/∂x not only in one direction (x), but also a phase gradient ∂Φ(x, y)/∂y in a direction (y) perpendicular to the x-direction, which is acquired by an additional phase stepping along the y-direction.
Another grating-based method to determine phase gradients ∂Φ(x, y)/∂x and/or ∂Φ(x, y)/∂y is proposed as “coded aperture” technique by A. Olivo and R. Speller (in: Phys. Med. Biol. 52, pp. 6555-6573 (2007)). In this method, a structured X-ray absorbing mask, named as “coded aperture”, is placed directly in front of an object to be examined. This mask provides small apertures, similar to the grating G0 in the Talbot-Lau type interferometer, which produce in case of a distant X-ray illumination an array of nearly parallel X-ray beams with a cross-sectional area, which is defined by the shape of the apertures. Therefore, a second coded aperture grating is placed directly in front of the detector. The apertures of the second mask are chosen such that a defined part of each X-ray beam is blocked, such that for each beam a predefined intensity is transmitted through the second mask. The detection unit behind the second mask is adapted such that the transmitted intensity of each X-ray beam is averaged and attached to one image pixel, respectively. As an object is put between the first and the second mask, each of said X-ray beams is diffracted by the object's structure, which results in an angular deflection of the beam as compared to the direction the beam would take in the absence of the object. The deflection of each beam causes a deviating signal for each beam in the detection unit, because the illumination area on the second mask is laterally translated. As a result, the signal acquired for each beam (meaning for each pixel) is proportional to the deflection of each beam, which therefore is proportional to a phase gradient ∂Φ(x, y)/∂x if the masks consists of parallel lines in y-direction. It is also possible to determine the phase gradients ∂Φ(x, y)/∂x and ∂Φ(x, y)/∂y independently if two-dimensional patterned coded apertures are used, and the direction of each beam deflection is determined by an additional stepping and sampling process, for example by translating the second mask into the x- and y-directions and therefore determining the x-component and the y-component of the phase gradient vector projected onto the detector plane.